IntermediateTopic 68 of 101Signals, Control & Comms
Lyapunov stability
Uses energy-like functions to prove stability without solving equations.
In depth
If a Lyapunov function V(x)>0 with V̇≤0 exists, equilibrium is stable; V̇<0 gives asymptotic stability. For linear systems V=xᵀPx with AᵀP+PA=-Q. Energy-like functions prove stability without solving ODEs, and design nonlinear controllers (backstepping).
Key points
- Positive definite functions
- LaSalle's invariance principle
- Control Lyapunov functions
ExampleTotal mechanical energy is a Lyapunov function for a damped pendulum.
What is Lyapunov stability?
Uses energy-like functions to prove stability without solving equations.
Why is it important to understand lyapunov stability?
Lyapunov methods prove stability without solving the nonlinear equations, which is valuable when exact solutions do not exist.
What is a Lyapunov function?
A scalar function that is positive away from equilibrium and decreases along trajectories, which shows the state is drawn to the equilibrium.
Is failing to find a Lyapunov function proof of instability?
No. Failing to find one does not prove anything, so other tests or a different candidate function may be needed.
How is Lyapunov stability used in practice?
Typical applications include proving that a pendulum with damping returns to rest, checking that an adaptive controller remains stable under all plausible conditions and designing a control law that makes an energy function decrease over time.
What should I learn before Lyapunov stability?
Study "Linearisation" first (topic 67 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
What are the key points of Lyapunov stability?
Positive definite functions. LaSalle's invariance principle. Control Lyapunov functions.
Can you give an example of Lyapunov stability?
Total mechanical energy is a Lyapunov function for a damped pendulum.
What comes after Lyapunov stability?
The next topic is "Model predictive control", which builds directly on this one.
How difficult is Lyapunov stability?
It is classed as Intermediate in Signals, Control & Comms (topic 68 of 101). It is an intermediate topic. Work through the beginner topics in Signals, Control & Comms first.